Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
Ecommerce: Agents and Rational Behavior
Lecture 8: Decision-Making under UncertaintyComplex Decisions
Ralf MöllerHamburg University of Technology
Literature
• Chapter 17
Material from Lise Getoor, Jean-ClaudeLatombe, Daphne Koller, and Russell
Sequential Decision Making
• Finite Horizon• Infinite Horizon
Simple Robot Navigation ProblemSimple Robot Navigation Problem
• In each state, the possible actions are U, D, R, and L
Probabilistic Transition ModelProbabilistic Transition Model
• In each state, the possible actions are U, D, R, and L• The effect of U is as follows (transition model):
• With probability 0.8 the robot moves up one square (if the robot is already in the top row, then it does not move)
Probabilistic Transition ModelProbabilistic Transition Model
• In each state, the possible actions are U, D, R, and L• The effect of U is as follows (transition model):
• With probability 0.8 the robot moves up one square (if the robot is already in the top row, then it does not move)• With probability 0.1 the robot moves right one square (if the robot is already in the rightmost row, then it does not move)
Probabilistic Transition ModelProbabilistic Transition Model
• In each state, the possible actions are U, D, R, and L• The effect of U is as follows (transition model):
• With probability 0.8 the robot moves up one square (if the robot is already in the top row, then it does not move)• With probability 0.1 the robot moves right one square (if the robot is already in the rightmost row, then it does not move)• With probability 0.1 the robot moves left one square (if the robot is already in the leftmost row, then it does not move)
Markov PropertyMarkov Property
The transition properties depend only on the current state, not on previous history (how that state was reached)
Sequence of ActionsSequence of Actions
• Planned sequence of actions: (U, R)
2
3
1
4321
[3,2]
Sequence of ActionsSequence of Actions
• Planned sequence of actions: (U, R)• U is executed
2
3
1
4321
[3,2]
[4,2][3,3][3,2]
HistoriesHistories
• Planned sequence of actions: (U, R)• U has been executed• R is executed
• There are 9 possible sequences of states – called histories – and 6 possible final states for the robot!
4321
2
3
1
[3,2]
[4,2][3,3][3,2]
[3,3][3,2] [4,1] [4,2] [4,3][3,1]
Probability of Reaching the GoalProbability of Reaching the Goal
•P([4,3] | (U,R).[3,2]) = P([4,3] | R.[3,3]) x P([3,3] | U.[3,2]) + P([4,3] | R.[4,2]) x P([4,2] | U.[3,2])
2
3
1
4321
Note importance of Markov property in this derivation
•P([3,3] | U.[3,2]) = 0.8•P([4,2] | U.[3,2]) = 0.1
•P([4,3] | R.[3,3]) = 0.8•P([4,3] | R.[4,2]) = 0.1
•P([4,3] | (U,R).[3,2]) = 0.65
Utility FunctionUtility Function
• [4,3] provides power supply• [4,2] is a sand area from which the robot cannot escape
-1
+1
2
3
1
4321
Utility FunctionUtility Function
• [4,3] provides power supply• [4,2] is a sand area from which the robot cannot escape• The robot needs to recharge its batteries
-1
+1
2
3
1
4321
Utility FunctionUtility Function
• [4,3] provides power supply• [4,2] is a sand area from which the robot cannot escape• The robot needs to recharge its batteries• [4,3] or [4,2] are terminal states
-1
+1
2
3
1
4321
Utility of a HistoryUtility of a History
• [4,3] provides power supply• [4,2] is a sand area from which the robot cannot escape• The robot needs to recharge its batteries• [4,3] or [4,2] are terminal states• The utility of a history is defined by the utility of the last state (+1 or –1) minus n/25, where n is the number of moves
-1
+1
2
3
1
4321
Utility of an Action SequenceUtility of an Action Sequence
-1
+1
• Consider the action sequence (U,R) from [3,2]
2
3
1
4321
Utility of an Action SequenceUtility of an Action Sequence
-1
+1
• Consider the action sequence (U,R) from [3,2]• A run produces one among 7 possible histories, each with some probability
2
3
1
4321
[3,2]
[4,2][3,3][3,2]
[3,3][3,2] [4,1] [4,2] [4,3][3,1]
Utility of an Action SequenceUtility of an Action Sequence
-1
+1
• Consider the action sequence (U,R) from [3,2]• A run produces one among 7 possible histories, each with some probability• The utility of the sequence is the expected utility of the histories:
U = ΣhUh P(h)
2
3
1
4321
[3,2]
[4,2][3,3][3,2]
[3,3][3,2] [4,1] [4,2] [4,3][3,1]
Optimal Action SequenceOptimal Action Sequence
-1
+1
• Consider the action sequence (U,R) from [3,2]• A run produces one among 7 possible histories, each with some probability• The utility of the sequence is the expected utility of the histories• The optimal sequence is the one with maximal utility
2
3
1
4321
[3,2]
[4,2][3,3][3,2]
[3,3][3,2] [4,1] [4,2] [4,3][3,1]
Optimal Action SequenceOptimal Action Sequence
-1
+1
• Consider the action sequence (U,R) from [3,2]• A run produces one among 7 possible histories, each with some probability• The utility of the sequence is the expected utility of the histories• The optimal sequence is the one with maximal utility• But is the optimal action sequence what we want to compute?
2
3
1
4321
[3,2]
[4,2][3,3][3,2]
[3,3][3,2] [4,1] [4,2] [4,3][3,1]
only if the sequence is executed blindly!
Accessible orobservable state
Repeat: s sensed state If s is terminal then exit a choose action (given s) Perform a
Reactive Agent AlgorithmReactive Agent Algorithm
Policy Policy (Reactive/Closed-Loop Strategy)(Reactive/Closed-Loop Strategy)
• A policy Π is a complete mapping from states to actions
-1
+1
2
3
1
4321
Repeat: s sensed state If s is terminal then exit a Π(s) Perform a
Reactive Agent AlgorithmReactive Agent Algorithm
Optimal PolicyOptimal Policy
-1
+1
• A policy Π is a complete mapping from states to actions• The optimal policy Π* is the one that always yields a history (ending at a terminal state) with maximal expected utility
2
3
1
4321
Makes sense because of Markov property
Note that [3,2] is a “dangerous” state that the optimal policy
tries to avoid
Optimal PolicyOptimal Policy
-1
+1
• A policy Π is a complete mapping from states to actions• The optimal policy Π* is the one that always yields a history with maximal expected utility
2
3
1
4321
This problem is called aMarkov Decision Problem (MDP)
How to compute Π*?
Additive UtilityAdditive Utility
• History H = (s0,s1,…,sn)• The utility of H is additive iff:
U(s0,s1,…,sn) = R(0) + U(s1,…,sn) = Σ R(i)
Reward
Additive UtilityAdditive Utility
• History H = (s0,s1,…,sn)• The utility of H is additive iff:
U(s0,s1,…,sn) = R(0) + U(s1,…,sn) = Σ R(i)
• Robot navigation example: R(n) = +1 if sn = [4,3]
R(n) = -1 if sn = [4,2]
R(i) = -1/25 if i = 0, …, n-1
Principle of Max Expected UtilityPrinciple of Max Expected Utility
• History H = (s0,s1,…,sn)• Utility of H: U(s0,s1,…,sn) = Σ R(i)
First-step analysis
• U(i) = R(i) + maxa ΣkP(k | a.i) U(k)
• Π*(i) = arg maxa ΣkP(k | a.i) U(k)
-1
+1
Value IterationValue Iteration
• Initialize the utility of each non-terminal state si toU0(i) = 0
• For t = 0, 1, 2, …, do: Ut+1(i) R(i) + maxa ΣkP(k | a.i) Ut(k)
-1
+1
2
3
1
4321
Value IterationValue Iteration
• Initialize the utility of each non-terminal state si toU0(i) = 0
• For t = 0, 1, 2, …, do: Ut+1(i) R(i) + maxa ΣkP(k | a.i) Ut(k)
Ut([3,1])
t0 302010
0.6110.5
0-1
+1
2
3
1
4321
0.705 0.655 0.3880.611
0.762
0.812 0.868 0.918
0.660
Note the importanceof terminal states andconnectivity of thestate-transition graph
Policy IterationPolicy Iteration
• Pick a policy Π at random
Policy IterationPolicy Iteration
• Pick a policy Π at random• Repeat:
Compute the utility of each state for Π Ut+1(i) R(i) + ΣkP(k | Π(i).i) Ut(k)
Policy IterationPolicy Iteration
• Pick a policy Π at random• Repeat:
Compute the utility of each state for Π Ut+1(i) R(i) + ΣkP(k | Π(i).i) Ut(k)
Compute the policy Π’ given theseutilities Π’(i) = arg maxa ΣkP(k | a.i) U(k)
Policy IterationPolicy Iteration
• Pick a policy Π at random• Repeat:
Compute the utility of each state for Π Ut+1(i) R(i) + ΣkP(k | Π(i).i) Ut(k)
Compute the policy Π’ given theseutilities Π’(i) = arg maxa ΣkP(k | a.i) U(k)
If Π’ = Π then return Π
Or solve the set of linear equations:
U(i) = R(i) + ΣkP(k | Π(i).i) U(k)
(often a sparse system)
n-Step n-Step decision processdecision process
Assume that:• Each state reached after n steps is terminal, hence has
known utility• There is a single initial state• Any two states reached after i and j steps are different
0 1 2 3
n-Step n-Step Decision ProcessDecision Process
For j = n-1, n-2, …, 0 do:For every state si attained after step j
Compute the utility of si
Label that state with the corresponding action
Π*(i) = arg maxa ΣkP(k | a.i) U(k)
U(i) = R(i) + maxa ΣkP(k | a.i) U(k)
0 1 2 3
What is the Difference?What is the Difference?
Π*(i) = arg maxa ΣkP(k | a.i) U(k)
U(i) = R(i) + maxa ΣkP(k | a.i) U(k)
0 1 2 3
-1
+1
Infinite HorizonInfinite Horizon
-1
+1
2
3
1
4321
In many problems, e.g., the robot navigation example, histories are potentially unbounded and the same state can be reached many timesOne trick:
Use discounting to make infiniteHorizon problem mathematicallytractable
What if the robot lives forever?
Example: Tracking a TargetExample: Tracking a Target
targetrobot
• The robot must keep the target in view• The target’s trajectory is not known in advance• The environment may or may not be known
An optimal policy cannot be computed aheadof time:- The environment might be unknown- The environment may only be partially observable- The target may not wait
A policy must be computed “on-the-fly”
POMDP POMDP (Partially Observable Markov Decision Problem)(Partially Observable Markov Decision Problem)
• A sensing operation returns multiple states, with a probability distribution
• Choosing the action that maximizes the expected utility of this state distribution assuming “state utilities” computed as above is not good enough, and actually does not make sense (is not rational)
Example: Target TrackingExample: Target Tracking
There is uncertaintyin the robot’s and target’s positions; this uncertaintygrows with further motion
There is a risk that the target may escape behind the corner, requiring the robot to move appropriately
But there is a positioninglandmark nearby. Shouldthe robot try to reduce itsposition uncertainty?
SummarySummary
• Decision making under uncertainty• Utility function• Optimal policy• Maximal expected utility• Value iteration• Policy iteration